Elsevier

Physics Letters A

Volume 375, Issue 23, 6 June 2011, Pages 2230-2233
Physics Letters A

Localization of hidden Chuaʼs attractors

https://doi.org/10.1016/j.physleta.2011.04.037Get rights and content

Abstract

The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chuaʼs circuit it is suggested to use a special analytical–numerical algorithm.

Highlights

► There are hidden attractors: basin doesnʼt contain neighborhoods of equilibria. ► Hidden attractors cannot be reached by trajectory from neighborhoods of equilibria. ► We suggested special procedure for localization of hidden attractors. ► We discovered hidden attractor in Chuaʼs system, L. Chua in his work didnʼt expect this.

Introduction

The classical attractors of Lorenz [1], Rossler [2], Chua [3], Chen [4], and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. The simplest examples of systems with such attractors are counterexamples to widely-known Aizermanʼs and Kalmanʼs conjectures on absolute stability (see, e.g., [8], [10]). Numerical localization, computation, and analytical investigation of such attractors are much more difficult problems. In the present Letter for localization of hidden attractors of Chuaʼs circuit it is suggested to use a special analytical–numerical algorithm.

Chuaʼs circuit can be described by differential equations in dimensionless coordinates:x˙=α(yx)αf(x),y˙=xy+z,z˙=(βy+γz). Here the functionf(x)=m1x+(m0m1)sat(x)=m1x+12(m0m1)(|x+1||x1|) characterizes a nonlinear element, of the system, called Chuaʼs diode; α, β, γ, m0, m1 are parameters of the system. In this system it was discovered the strange attractors [11], [12] called then Chuaʼs attractors (for the current state of chaotic behavior investigation in Chuaʼs circuit see, e.g., recent work [13] and references within).

To date all known Chuaʼs attractors are the attractors that are excited from unstable equilibria. This makes it possible to compute different Chuaʼs attractors [14], [15], [16] with relative easy.

The applied in this Letter algorithm shows for the first time the possibility of existence of hidden attractor in system (1). Note that L. Chua himself, analyzing in the work [3] different cases of attractor existence in Chuaʼs circuit, does not admit the existence of such hidden attractor.

Section snippets

Analytical–numerical method for attractors localization

Consider a system with one scalar1 nonlinearitydxdt=Px+qψ(rx),xRn. Here P is a constant (n×n)-matrix, q,r are constant n-dimensional vectors, is a transposition operation, ψ(σ) is a continuous piecewise-differentiable2 scalar function, and ψ(0)=0. Define a coefficient of harmonic linearization k in

Localization of hidden attractor in Chuaʼs system

We now apply the above algorithm to analysis of Chuaʼs system. For this purpose, rewrite Chuaʼs system (1) in the form (3)dxdt=Px+qψ(rx),xR3. HereP=(α(m1+1)α01110βγ),q=(α00),r=(100),ψ(σ)=(m0m1)sat(σ).

Introduce the coefficient k and small parameter ε, and represent system (12) as (6)dxdt=P0x+qεφ(rx), whereP0=P+kqr=(α(m1+1+k)α01110βγ),λ1,2P0=±iω0,λ3P0=d,φ(σ)=ψ(σ)kσ=(m0m1)sat(σ)kσ. By nonsingular linear transformation x=Sy system (13) is reduced to the form (8)dydt=Ay+bεφ(cy),

Conclusions

In the present Letter the application of special analytical–numerical algorithm for hidden attractor localization is discussed and the existence of such hidden attractor in Chuaʼs circuits is demonstrated.

Acknowledgements

This work was supported by the Academy of Finland, the Ministry of Education and Science (Russia), and Saint-Petersburg State University.

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